Shaft Torque, Speed, and Power Relationship
Converts rotational velocity and torque load into equivalent mechanical power, and vice-versa, in standard units.
Primary Mathematical Expression
Nomenclature & Variables
| Symbol | Variable Name | Metric Unit | Imperial Unit | Description |
|---|---|---|---|---|
| P | Power | kW | HP | The rate of mechanical work output. |
| T | Torque | N·m | ft·lb | The rotational force moment transmitted. |
| n | Speed | rpm | rpm | The rotational velocity of the shaft. |
Step-by-Step Derivation
- 1
Work done by a constant torque per revolution is W = 2 * \pi * T.
- 2
For n revolutions per minute, the work per second (Power) is: P = W * (n / 60) = (2 * \pi * T * n) / 60 = T * n * (\pi / 30).
- 3
In SI units where Power is in Watts (W) and Torque is in N·m: P_W = T * n * 0.10472.
- 4
Converting Power to kilowatts (1 kW = 1000 W): P_kW = T * n * 0.10472 / 1000 = (T * n) / 9549.3.
- 5
Rounding the divisor to 9550 gives the standard mechanical engineering equation: P = T * n / 9550.
Worked Example Calculation
An electric motor rotates at 1450 rpm and transmits a torque of 180 N·m. Calculate the output power in kilowatts.
- •Identify input parameters: T = 180 N·m, n = 1450 rpm.
- •Apply the Torque-Power Relation Formula: P = T * n / 9550.
- •Compute: P = (180 * 1450) / 9550 = 261,000 / 9550 = 27.33 kW.
Engineering Assumptions
- •Constant torque and steady speed over the calculation interval.
- •System behaves as a rigid rotating assembly with no slip.
Design Limitations
- •Divisor 9550 is specific to SI units (kilowatts, N·m, and rpm). Metric/Imperial conversion factors must change if using hp and ft·lb.
Academic References & Standards
Shigley's Mechanical Engineering Design, 11th Edition
Textbook covering transmission power and gear torque relationships.